Theorem resco 5115

Description: Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.)

Assertion

Ref	Expression
resco	|- ( ( A o. B ) |` C ) = ( A o. ( B |` C ) )

Proof of Theorem resco

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relres 4919	. 2 |- Rel ( ( A o. B ) |` C )
2		relco 5109	. 2 |- Rel ( A o. ( B |` C ) )
3		vex 2733	. . . . . 6 |- x e. _V
4		vex 2733	. . . . . 6 |- y e. _V
5	3, 4	brco 4782	. . . . 5 |- ( x ( A o. B ) y <-> E. z ( x B z /\ z A y ) )
6	5	anbi1i 455	. . . 4 |- ( ( x ( A o. B ) y /\ x e. C ) <-> ( E. z ( x B z /\ z A y ) /\ x e. C ) )
7		19.41v 1895	. . . 4 |- ( E. z ( ( x B z /\ z A y ) /\ x e. C ) <-> ( E. z ( x B z /\ z A y ) /\ x e. C ) )
8		an32 557	. . . . . 6 |- ( ( ( x B z /\ z A y ) /\ x e. C ) <-> ( ( x B z /\ x e. C ) /\ z A y ) )
9		vex 2733	. . . . . . . 8 |- z e. _V
10	9	brres 4897	. . . . . . 7 |- ( x ( B |` C ) z <-> ( x B z /\ x e. C ) )
11	10	anbi1i 455	. . . . . 6 |- ( ( x ( B |` C ) z /\ z A y ) <-> ( ( x B z /\ x e. C ) /\ z A y ) )
12	8, 11	bitr4i 186	. . . . 5 |- ( ( ( x B z /\ z A y ) /\ x e. C ) <-> ( x ( B |` C ) z /\ z A y ) )
13	12	exbii 1598	. . . 4 |- ( E. z ( ( x B z /\ z A y ) /\ x e. C ) <-> E. z ( x ( B |` C ) z /\ z A y ) )
14	6, 7, 13	3bitr2i 207	. . 3 |- ( ( x ( A o. B ) y /\ x e. C ) <-> E. z ( x ( B |` C ) z /\ z A y ) )
15	4	brres 4897	. . 3 |- ( x ( ( A o. B ) |` C ) y <-> ( x ( A o. B ) y /\ x e. C ) )
16	3, 4	brco 4782	. . 3 |- ( x ( A o. ( B |` C ) ) y <-> E. z ( x ( B |` C ) z /\ z A y ) )
17	14, 15, 16	3bitr4i 211	. 2 |- ( x ( ( A o. B ) |` C ) y <-> x ( A o. ( B |` C ) ) y )
18	1, 2, 17	eqbrriv 4706	1 |- ( ( A o. B ) |` C ) = ( A o. ( B |` C ) )

Syntax hints:   /\ wa 103   = wceq 1348   E.wex 1485   e. wcel 2141   class class class wbr 3989   |` cres 4613   o. ccom 4615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-co 4620  df-res 4623

This theorem is referenced by:  cocnvcnv2  5122  coires1  5128  relcoi1  5142  dftpos2  6240